Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution

The widely used Cox proportional hazards regression model for the analysis of censored survival data has limited utility when either hazard functions themselves are of primary interest, or when relative times instead of relative hazards are the relevant measures of association. Parametric regression models are an attractive option in situations such as this, although the choice of a particular model from the available families of distributions can be problematic. The generalized gamma (GG) distribution is an extensive family that contains nearly all of the most commonly used distributions, including the exponential, Weibull, log normal and gamma. More importantly, the GG family includes all four of the most common types of hazard function: monotonically increasing and decreasing, as well as bathtub and arc-shaped hazards. We present here a taxonomy of the hazard functions of the GG family, which includes various special distributions and allows depiction of effects of exposures on hazard functions. We applied the proposed taxonomy to study survival after a diagnosis of clinical AIDS during different eras of HIV therapy, where proportionality of hazard functions was clearly not fulfilled and flexibility in estimating hazards with very different shapes was needed. Comparisons of survival after AIDS in different eras of therapy are presented in terms of both relative times and relative hazards. Standard errors for these and other derived quantities are computed using the delta method and checked using the bootstrap. Description of standard statistical software (Stata, SAS and S-Plus) for the computations is included and available at http://statepi.jhsph.edu/software.